On some subclasses of m-fold symmetric bi-univalent functions

Lemma 1 [14] If p (z) = 1 + p1z + p2z2 + p3z3 + · · · is an analytic function in U with positive real part, then |pn| ≤2 (n ∈N = {1, 2, . . .}) and p2 −p2 1 2 ≤2 −|p1|2 2 . 2 Coefficient bounds for the function class SΣm(α, λ) Definition 2 A function f ∈Σm is said to be in the class SΣm(α, λ) if the following conditions are satisfied:

Theorem 3 Let f given by (4) be in the class SΣm(α, λ), 0 < α ≤1. Then |am+1| ≤ 4λα m p (1 + λ) [4λα + (1 + λ)(1 −α)] + 2α(1 −λ) and |a2m+1| ≤ 2λα m (1 + λ) + 8(m + 1)λ2α2 m2(1 + λ)2 .

Theorem 5 Let f given by (4) be in the class SΣm(β, λ), 0 ≤β < 1. Then |am+1| ≤2λ m s 2 (1 −β) 2λ2 + λ + 1 and |a2m+1| ≤8(m + 1)λ2 (1 −β)2 m2(1 + λ)2 + 2λ (1 −β) m (1 + λ) . 5

Corollary 6 ( see [4]) Let f given by (4) be in the class Sα Σm (0 < α ≤1). Then |am+1| ≤ 2α m√α + 1 and |a2m+1| ≤α m + 2(m + 1)α2 m2 .

Corollary 7 ( see [4]) Let f given by (4) be in the class Sβ Σm (0 ≤β < 1). Then |am+1| ≤ p 2 (1 −β) m and |a2m+1| ≤2(m + 1)(1 −β)2 m2 + 1 −β m . The classes Sα

Theorem 5 reduce to Corollary 10 and Corollary 11, respectively, which were proven earlier by Murugunsundaramoorthy et al. [12].

Corollary 10 Let f given by (4) be in the class S∗ Σ(α) (0 < α ≤1). Then |a2| ≤ 2α √α + 1 and |a3| ≤4α2 + α.

Corollary 11 Let f given by (4) be in the class S∗ Σ(β) (0 ≤β < 1). Then |a2| ≤ p 2 (1 −β) and |a3| ≤4(1 −β)2 + (1 −β). References [1] Altınkaya, S¸., Yal¸cın, S.: Initial coefficient bounds for a general class of bi- univalent functions, International Journal of Analysis, Article ID 867871, 4 pp, (2014). [2] Altınkaya, S¸., Yal¸cın, S.: Coefficient Estimates for Two New Subclasses of Bi-univalent Functions with respect to Symmetric Points, Journal of Function Spaces, Article ID 145242,5 pp, (2015).